You need to deploy openssl libraries with your app. This can be done by putting appropiate dlls (ssleay32.dll and few more) alongside your executable or by installing openssl runtime package (eg. from here: "Link":http://www.openssl.org/related/binaries.html ) on users computer.

Well this is clearly the issue with missing openssl libraries. Are you sure that you've got the same openssl libs version as what your app uses? The easiest way to find out would be to explicitly put libeay32.dll and ssleay32.dll in your app executable directory and see if it works.
Also - is your Qt build with openssl support enabled?